 Question Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules | Nagwa Question Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules | Nagwa

# Question Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules Mathematics

Find the first derivative of the function 𝑦 = −7𝑥⁴ ln 6𝑥⁴.

02:12

### Video Transcript

Find the first derivative of the function 𝑦 equals negative seven 𝑥 to the fourth power times the natural log of six 𝑥 to the fourth power.

Here, we have the product of two differentiable functions. The first is negative seven 𝑥 to the fourth power and the second is the natural log of six 𝑥 of the fourth power. We can, therefore, find the first derivative of our function by using the product rule. This says that the derivative of the product of two differentiable functions, 𝑢 and 𝑣, is 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥. If we let 𝑢 be equal to negative seven 𝑥 to the fourth power, then d𝑢 by d𝑥 is equal to four times negative seven 𝑥 cubed. That’s negative 28𝑥 cubed. We then let 𝑣 be equal to the natural log of six 𝑥 to the fourth power.

So how do we obtain d𝑣 by d𝑥? Well, we can do one of two things. We could spot that we have a composite function and use the chain rule to find its derivative. Alternatively, we can quote the general result for the derivative of the logarithm of some function 𝑓 of 𝑥. This says that if 𝑦 is equal to the natural log of this function 𝑓 of 𝑥, then d𝑦 by d𝑥 is equal to 𝑓 prime of 𝑥, the derivative of that function, divided by the original function 𝑓 of 𝑥. In this case, 𝑓 of 𝑥 is equal to six 𝑥 to the fourth power. So 𝑓 prime of 𝑥, the derivative of this, is four times six 𝑥 cubed, which is 24𝑥 cubed. d𝑣 by d𝑥 is, therefore, 24𝑥 cubed, divided by the original function, six 𝑥 to the fourth power.

Well, this simplifies quite nicely to four over 𝑥. And we can now substitute everything we have into the formula for the product rule. d𝑦 by d𝑥 is 𝑢 times d𝑣 by d𝑥. That’s negative seven 𝑥 to the fourth power times four over 𝑥 plus 𝑣 times d𝑢 by d𝑥. That’s the natural log of six 𝑥 to the fourth power times negative 28𝑥 cubed. We simplify, and then we factor negative 28𝑥 cubed. And we obtained d𝑦 by d𝑥 to be negative 28𝑥 cubed times the natural logarithm six 𝑥 to the fourth power plus one.